1550251515-Classical_Complex_Analysis__Gonzalez_

134

v.

LI

Fig. 3.6

Alternatively, if for every e > 0 there is a N' ( oo) such that

zEN'(oo)nD

(Fig. 3.6). We write

Z->00 lim f(z) = L or

implies f(z) E N,(L)

f(z)--+L as z--+ oo

Chapter 3

By replacing the z-plane by the corresponding Riemann sphere (Fig.
3.7), and the Euclidean metric by the chordal metric, an e-8 type of def-
inition, similar to Definition 3.1, can be stated for this case, as follows:
limz-+oo f(z) = L if for every € > 0 there is a 8 > 0 such that

lf(z)-Ll<e whenever 0 < x(z,oo) < 8 and z ED

If the limit of f(z) at oo exists, it is unique and the proof is similar to

that of Theorem 3.1.

0

Fig. 3.7